Bottom Line:
The circular spatial scan statistics shows a high level of accuracy in detecting circular clusters exactly.The proposed spatial scan statistic is shown to work well for small to moderate cluster size, up to say 30.For larger cluster sizes, the method is not practically feasible and a more efficient algorithm is needed.

Background: The spatial scan statistic proposed by Kulldorff has been applied to a wide variety of epidemiological studies for cluster detection. This scan statistic, however, uses a circular window to define the potential cluster areas and thus has difficulty in correctly detecting actual noncircular clusters. A recent proposal by Duczmal and Assunção for detecting noncircular clusters is shown to detect a cluster of very irregular shape that is much larger than the true cluster in our experiences.

Methods: We propose a flexibly shaped spatial scan statistic that can detect irregular shaped clusters within relatively small neighborhoods of each region. The performance of the proposed spatial scan statistic is compared to that of Kulldorff's circular spatial scan statistic with Monte Carlo simulation by considering several circular and noncircular hot-spot cluster models. For comparison, we also propose a new bivariate power distribution classified by the number of regions detected as the most likely cluster and the number of hot-spot regions included in the most likely cluster.

Results: The circular spatial scan statistics shows a high level of accuracy in detecting circular clusters exactly. The proposed spatial scan statistic is shown to have good usual powers plus the ability to detect the noncircular hot-spot clusters more accurately than the circular one.

Conclusion: The proposed spatial scan statistic is shown to work well for small to moderate cluster size, up to say 30. For larger cluster sizes, the method is not practically feasible and a more efficient algorithm is needed.

Figure 3: The most likely cluster detected by the circular and the flexible spatial scan statistic. (a) Detected by the circular spatial scan statistic for both K = 15 and K = 20 and (b) by the flexible spatial scan statistic for both K = 15 and K = 20, when applied to a random sample from the cluster model C = {14, 15, 26, 27}.

Mentions:
Irrespective of the value of K, the circular spatial scan statistic detected the regions {14, 15} as MLC with log likelihood ratio = 20.1, p = 1/(999 + 1) = 0.001 and the estimated relative risk is = 3.47. This is shown in Figure 3(a). The flexible spatial scan statistic, regardless of the value K, detected the regions {14, 15, 26, 27, 33} as MLC with log likelihood ratio = 29.7, p = 0.001 and the estimated relative risk is = 3.41. This is shown in Figure 3(b). Duczmal and Assunção's method, on the other hand, detected a cluster of peculiar shape that is much larger than the true cluster. In the case of K = 15, their scan statistic detected an area consisting of K = 15 connected regions {14, 15, 24, 26, 27, 31, 32, 33, 48, 54, 69, 77, 78, 90, 110 } as MLC with log likelihood ratio = 31.8, p = 0.001 and the estimated relative risk is = 2.40. This is shown in Figure 4(a). Figure 4(b) shows the most likely cluster {14, 15, 26, 27, 31, 32, 33, 48, 60, 61, 62, 67, 69, 77, 78, 80, 89, 90, 108, 110 } detected by Duczmal and Assunção's scan statistic for K = 20 where the length of MLC is also the same as K = 20 and log likelihood ratio = 36.0, p = 0.001 and the estimated relative risk is = 2.26. In the case of K = 15, the results of the three scan statistics are summarized in Table 1. Although the most likely cluster detected by Duczmal and Assunção's scan statistic has the largest log likelihood ratio among three scan statistics, it has detected MLC surprisingly larger than the true cluster.

Figure 3: The most likely cluster detected by the circular and the flexible spatial scan statistic. (a) Detected by the circular spatial scan statistic for both K = 15 and K = 20 and (b) by the flexible spatial scan statistic for both K = 15 and K = 20, when applied to a random sample from the cluster model C = {14, 15, 26, 27}.

Mentions:
Irrespective of the value of K, the circular spatial scan statistic detected the regions {14, 15} as MLC with log likelihood ratio = 20.1, p = 1/(999 + 1) = 0.001 and the estimated relative risk is = 3.47. This is shown in Figure 3(a). The flexible spatial scan statistic, regardless of the value K, detected the regions {14, 15, 26, 27, 33} as MLC with log likelihood ratio = 29.7, p = 0.001 and the estimated relative risk is = 3.41. This is shown in Figure 3(b). Duczmal and Assunção's method, on the other hand, detected a cluster of peculiar shape that is much larger than the true cluster. In the case of K = 15, their scan statistic detected an area consisting of K = 15 connected regions {14, 15, 24, 26, 27, 31, 32, 33, 48, 54, 69, 77, 78, 90, 110 } as MLC with log likelihood ratio = 31.8, p = 0.001 and the estimated relative risk is = 2.40. This is shown in Figure 4(a). Figure 4(b) shows the most likely cluster {14, 15, 26, 27, 31, 32, 33, 48, 60, 61, 62, 67, 69, 77, 78, 80, 89, 90, 108, 110 } detected by Duczmal and Assunção's scan statistic for K = 20 where the length of MLC is also the same as K = 20 and log likelihood ratio = 36.0, p = 0.001 and the estimated relative risk is = 2.26. In the case of K = 15, the results of the three scan statistics are summarized in Table 1. Although the most likely cluster detected by Duczmal and Assunção's scan statistic has the largest log likelihood ratio among three scan statistics, it has detected MLC surprisingly larger than the true cluster.

Bottom Line:
The circular spatial scan statistics shows a high level of accuracy in detecting circular clusters exactly.The proposed spatial scan statistic is shown to work well for small to moderate cluster size, up to say 30.For larger cluster sizes, the method is not practically feasible and a more efficient algorithm is needed.

Background: The spatial scan statistic proposed by Kulldorff has been applied to a wide variety of epidemiological studies for cluster detection. This scan statistic, however, uses a circular window to define the potential cluster areas and thus has difficulty in correctly detecting actual noncircular clusters. A recent proposal by Duczmal and Assunção for detecting noncircular clusters is shown to detect a cluster of very irregular shape that is much larger than the true cluster in our experiences.

Methods: We propose a flexibly shaped spatial scan statistic that can detect irregular shaped clusters within relatively small neighborhoods of each region. The performance of the proposed spatial scan statistic is compared to that of Kulldorff's circular spatial scan statistic with Monte Carlo simulation by considering several circular and noncircular hot-spot cluster models. For comparison, we also propose a new bivariate power distribution classified by the number of regions detected as the most likely cluster and the number of hot-spot regions included in the most likely cluster.

Results: The circular spatial scan statistics shows a high level of accuracy in detecting circular clusters exactly. The proposed spatial scan statistic is shown to have good usual powers plus the ability to detect the noncircular hot-spot clusters more accurately than the circular one.

Conclusion: The proposed spatial scan statistic is shown to work well for small to moderate cluster size, up to say 30. For larger cluster sizes, the method is not practically feasible and a more efficient algorithm is needed.